torsdag 21 april 2016

Reformulation of Clay Navier-Stokes Problem Needed 1

The formulation of the Clay Millennium Problem about global smoothness of solutions to the incompressible Navier-Stokes equations by Charles Fefferman, circumvents the phenomenon of turbulence as the most important aspect of fluid flow from both mathematical and physical point of view. The result is a problem which is both meaningless and without solution, and thus cannot serve well as a Clay Millennium Problem.

The unfortunate formulation by Fefferman comes out in the recent attempts by Terence Tao to construct a solution with local blow-up of fluid speed to infinity in finite time. Tao thus seeks a negative answer to global smoothness by constructing solutions with flow speed going to infinity locally. But he does not succeed and there is no reason to expect that he ever will, because the viscous term in Navier-Stokes dominates the convective term on small scales. Tao working in conjunction with Fefferman, thus is led into a fruitless direction.

The question of global smoothness in Fefferman's formulation, should better be replaced by a question of turbulence with turbulent flow defined as flow with velocity $u(x,t)$ with given initial data $u(x,0)$ such that for a positive constant $C$ (which is not small)
  • $\frac{\nu \int\vert\nabla u(x,t)\vert^2 dx}{\int \vert u(x,t)\vert^2 dx} > C$ for $t>0$    
for all small viscosities $\nu > 0$. Note that by renormalizing initial data, the effective value of the viscosity can always be made as small as desired, see PS below. Non-turbulent = laminar flow then has a constant $C$ which tends to zero with $\nu$. 

A turbulent solution would then correspond to a non-smooth solution in Fefferman's formulation, and then a laminar = non-turbulent solution to a smooth solution, and a Clay problem about global existence of laminar solutions would have a negative answer: For any given positive viscosity, there is data such that the corresponding Navier-Stokes solution becomes turbulent in finite time. Or turned the other way: For many initial data there is a viscosity such that the corresponding solution becomes turbulent in finite time.

Another aspect of Fefferman's unfortunate formulation is that the flow is supposed to fill all of space, or be periodic in space, which means that the completely crucial presence of flow boundary and choice of boundary condition, is neglected. There can be no rational reason to formulate a mathematical problem presented as being connected to physical reality of importance to humanity, in a way that makes any such connection meaningless.

PS1 renormalisation goes as follows: If $u(x,t)$ satisfies
  • $\frac{\partial u}{\partial t}+u\cdot\nabla u-\nu\Delta u = 0$, 
then $\bar u=\frac{u}{\alpha}$ with $\alpha >0$ satisfies
  • $\alpha\frac{\partial\bar u}{\partial t}+\bar u\cdot\nabla\bar u-\alpha\nu\Delta\bar u = 0$ 
and thus $\bar u$ with renormalisation of time $t=\alpha\bar t$ satisfies:
  • $\frac{\partial\bar u}{\partial\bar t}+\bar u\cdot\nabla\bar u-\bar\nu\Delta \bar u = 0$ 
with $\bar\nu =\bar\alpha\nu$ arbitrarily small with $\alpha >0$.

PS2 The Clay Navier Stokes problem is presented by the Clay Mathematical Institute as follows:
  • Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
But in the official formulation of the problem by Fefferman there is nothing about turbulence! Instead, it appears that the problem is deliberately cleverly formulated so as to completely exclude this fundamental aspect from the discussion, by focussing on blow-up instead of turbulence as expression of non-smoothness. I have many times tried to get this across to people spending time on seeking a solution and to mr Clay ready to spend money on a solution, so far with little success. But  for every year without solution the lack of meaning of the present problem formulation may become more understood. Maybe time is now ripe for a revision of the formulation of the problem, mr Clay?
In any case, waving with turbulence and then excluding turbulence is not correct science.

PS3 Here is copy of a letter to Tao:

Hi Terence

I see that you seek to construct solutions with blow-up to Euler/Navier-Stokes equations in an effort to solve the Clay Navier-Stokes problem as formulated by Fefferman. I have already tried to get across to you and Fefferman that the present formulation is not correct from scientific point of view (as I see it), since turbulence is named as the main unresolved mystery of the Navier-Stokes equations in the presentation of the problem by the Clay Institute, yet the formulation by Fefferman is made so as to exclude turbulence from the discussion.

I would appreciate if you could give your view on this apparent contradiction. I would also be happy if you could comment on the reformulation of the problem including turbulence suggested here:

I understand that you may want to discard my proposal because your time is limited, but working on a problem without meaningful answer may represent even more loss of time.

Best regards


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